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Lesson: _____ Section 6.3,5 Intro to Differential Equations & The Equations of Motion
An equation containing derivatives is called a βdifferential equation.β Ex. A car is driving in a straight line at a constant velocity of 50 mph. If s is the distance of the car from a fixed point, and t is time in hours, then This is called βrectilinear motionβ (straight line) This is called a βdifferential equationβ for function s. This says the derivative of s is 50. ππ ππ‘ =50 50ππ‘ The βsolutionβ to the diff. eq. is function s, which is the antiderivative of 50 π = =50π‘+π This is a general solution. To make it specific, we would need to know an initial value. In this case, when π‘=0 we know that s = c, therefore c represents the initial distance from the reference point.
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This diff. eq. represents the acceleration. Why is it negative?
Ex. An apple is dropped from a 100 ft building. Find position and velocity as a function of time. When does it hit the ground? How fast is is going when it hits? Acceleration due to gravity π=9.8 π π 2 ππ 32 ππ‘ π 2 If distance is in feet, and time is in seconds, thenβ¦ ππ£ ππ‘ =β32 This diff. eq. represents the acceleration. Why is it negative? π£=β32π‘+c Since π£=π when π‘=0 then c must represent our initial velocity. Letβs call it π£ 0 . π£=β32π‘+ π£ 0 Since the apple was dropped, π π = π£=β32π‘ π ππ ππ‘ Now, π£= =β32π‘ Letβs rewrite this function for velocity as a differential equation for position π =β16 π‘ 2 +π π =β16 π‘ 2 + π 0 Since π =π when π‘=0 then k must represent our initial position. Letβs call it π 0 . π =β16 π‘ We know that π π =πππ
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When does it hit the ground?
How fast is is going when it hits? Think about it. What does it mean in terms of position when the apple hits the ground? π=π Since π =β16 π‘ We want to solve the equation: =β16 π‘ 16 π‘ 2 =100 π‘ 2 = π‘=Β± =Β± 10 4 =Β± 2.5 The apple hits after 2.5 seconds This is asking for π£(2.5) π£=β32π‘ π£ 2.5 =β =β80 ππ‘ π Should this be negative? Yes! Our position is decreasing which means a negative rate of change. See also p. 295 example Note initial velocity is given. How do we find the βhighest pointβ using the derivative? Take a minute to READ 6.5 (history of the equations of motion)
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